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Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

Markos A. Katsoulakis·Petr Plechac·Luc Rey-Bellet·Dimitrios K. Tsagkarogiannis
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Abstract

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first-and often inadequate-approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.

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Markos A. Katsoulakis, Petr Plechac, Luc Rey-Bellet, Dimitrios K. Tsagkarogiannis. 2006-08-01. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. https://arxiv.org/abs/math/0608007

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